The design paradigm of many engineering systems (aircrafts, bridges, wind mills or offshore drilling risers) is that the structural compliance should be minimized to avoid large deformations induced by hydrodynamic loads. Dynamic structural deformations induced by fluid loads were first ignored, unfortunately leading to dramatic consequence. The accident of the Tacoma Narrows suspension bridge is a famous example of disasters due to a dynamic fluid/structure interaction, known as the flutter phenomenon. Such phenomenon is also well known in aeronautics, where large-amplitude deformations of the wings may occur when the airplane’s speed is increased beyond a critical speed, thus limiting the flight envelope. Although the flutter manifests through large-amplitude structural-deformation, it is due to a linear instability arising from the interaction between structural and fluid dynamics, which both exhibit stable dynamics in the absence of coupling. If flutter could be controlled at cruise speeds, not only the flight envelope could be extended, but also lighter wings could be designed, thus improving the energetic efficiency of airplanes. In the offshore marine industry, very elongated drilling risers are used to extract oil from the bed sea. Large displacements of the risers are observed as a result of large-scale vortices shed in the wake flow behind the risers. The vortex-induced deformation of the risers is also due to a linear instability arising from the interaction between structural and fluid dynamics, but with a different physical origin: the resonance between two physical oscillators characterized by very close natural frequencies. In all of these examples, the large displacements or deformations of the structures are induced by flow-induced elastic-deformation instabilities and must be avoided, since they are currently limiting the capacities of products in various industrial branches such as aeronautics, marine industry and wind electricity production. If suppressing flow-induced elastic instabilities is an ultimate goal in most today’s industrial applications, observation of nature strongly suggests that structural compliance can also be an advantage. A first striking example is the dynamic small-amplitude deformation of dolphin’s skin that helps them to reduce their skin-friction drag by delaying the laminar-turbulent transition process. A second example is the large-amplitude motion/deformation of appendices used by swimming animal (fishes and eels) or flying animals (insect or birds) to generate propulsive and lift forces, respectively. The flexibility of structures to increase the propulsive forces is already effective in nature. The objectives of the AEROFLEX are both physical and methodological. The two physical objectives are (i) the suppression of flow-induced elastic instabilities that are limiting the capacities of products in various industrial configurations and (ii) the use of elastic structures to reduce the skin-friction drag of blunt bodies and potentially generate mean propulsive forces at a low energetic cost. To achieve these physical objectives, innovative methodologies are developed. General mathematical formulations and numerical methods are developed to (i) determine flow-induced elastic instabilities in steady or oscillations configurations that are encountered in industrial or natural configurations, and (ii) control (suppress or use) these flow-induced instabilities with adjoint-based optimization of the structure.